# Functions and Length Scales

Regretfully I have to start with an apology as I fear I might be unable to express the question rigorously.

Often reading physics papers the concept of "length scale" is used, in statements such as "over this length scale, the phenomenon can be characterized by an exponential decay", or "the increase in X is virtually linear over such and such time scale". The Nobel Laureate De Gennes seems to me a virtuoso in this particular art.

I am able to follow some reasoning, but I am not so sure I understand fully their methods. For example, let us imagine I have got a model characterizing the change of a certain quantity, $Y$, versus time $t$: $$Y = e^{-A/t}$$ where $A$ is a constant. The function tends to zero for small times, it is first concave and then becomes convex. One expects to be able to characterize the "length scales" at which the transition occurs, in relation to the constant $A$. How can that be done? I tried to calculate the second derivative, which equals $$Y'' = e^{-A/t} \frac{A^2}{t^4} - 2 e^{-A/t} \frac{A}{t^3}$$ so imposing the conditions it equals zero I get the equation $$\frac{A}{t^4} = \frac {2}{t^3}$$ suggesting the conclusion the concave-convex transition occurs at "time scales in the order of A", is this correct?

But what truly puzzles me is how to characterize, for example, the time scale over which the functions is "almost flat", in the initial concave region. For example, if one fixes $A = 100$ and plot the function for a maximum $t = 5$, intuitively one suspects there must be a way to say "over such and such length scale, the function is flat". I wonder if to give sense to this statement one should specify which variations in the functions can be considered negligible.

Thanks a lot for any help on this I appreciate rather misty question.

• The 2nd derivative goes to zero at $t = A/2$, which tells you where the change in the slope (or 1st derivative) goes to zero, right? You can define a range of magnitudes of the 2nd derivative that would correspond to "small values," which would allow you to characterize the time scales over which your function is nearly flat. Jan 22, 2016 at 14:20
• The use of terms like "length scales" and "time scales" are ways to approximate a result as a method for determining when a term may be important or negligible. I am not sure at what level you are at in your studies, but the further you go in physics the more you will find that it really becomes, as one my grad school professors put it, "the art of approximation..." Jan 22, 2016 at 14:22
• I'm a little confused by your mention of a transition from concave to convex. $e^{-t/A}$ is concave "up" everywhere. Jan 22, 2016 at 14:29
• for the last part of your question : In a Monte Carlo NVT simulation How do I determine equilibration asks almost the same in the context of a simulation
– user46925
Jan 22, 2016 at 14:58
• As for your specific example, note that $A$ is the only number with units of time of your system. So if something is to be a time scale, it has to be $A$. In general, the units of the quantities can simplify the search for scales. Jan 22, 2016 at 18:19

The time scale for which the function is "almost flat" depends on the specifics problem. the most straightforward way would be to define "flat" as changing very slowly. how slow is slowly? well, that depends on your problem, and on all the other time scales involved.

For example: let's say you are on a ship, and there are waves in the sea. if the length of the wave is comparable with the length of ship, than the ship will be tilted and the wave is not "flat". if the length of the wave is very long compared with the boat, the boat will be tilted very slightly, so the wave can be considered as "flat".

If you had a smaller ship, the first wave could be considered flat! so It really depends on the length of the ship.

My point in this that the definition of "flat" depends on other scales in your problem. ask yourself what other important time scales are involved, and than you will know what "flat" is.

You are right in your understanding, and for the example you put is right to say that the transitions occurs in time scales of the order of A.

In general you talk about time scales when the scales are more relevant than the actual value, or when you are talking generally about a process for which the different values do not determine the physics.

For example the life of a fly is in the time scales of the lunar cycle, there you get the idea that in average they might live more than one week and less than for weeks, maybe a bit more.

In physics this is important because many phenomena are scale dependent, is important to know them accurately. For example neutrons behave very different at different scales of energy, but if you care for the values (linearly I mean) you would only see one type of behavior (depending on the scale you are) which does not show much difference in the physics.